Impact of Source Counter on Routing Performance in Resource Constrained DTNs

Transcription

1 Impact of Source Counter on Routing Performance in Resource Constrained DTNs Xiaolan Zhang a,, Honggang Zhang b, Yu Gu c a Dept. of Computer and Information Sciences, Fordham University b Dept. of Mathematics and Computer Science, Suffolk University, Boston, MA c NEC Laboratories, America, 4 Independence Way, Princeton, NJ Abstract We study routing schemes for Disruption Tolerant Networks (DTNs) where transmission bandwidth is scarce resource. In such a setting, a key issue is how to schedule the transmission of packets under limited bandwidth to optimize performance. Such a scheduling consists of source control (i.e., source nodes choosing a routing scheme) and the local transmission scheduling performed by each node. Existing works typically focus on transmission scheduling and buffer management aspects, but due to theoretical and practical difficulties, only heuristics have been proposed. In this work, we explore an alternative way to improve DTN routing performance via source control. We first show through simulation that for spray-and-wait routing scheme where the source node specifies the maximum allowed number of copies of a packet in the network, there exists an optimal counter value that achieves the minimum network-wide average packet delivery delay. Then as a first step towards understanding multi-hop multi-copy DTN routing schemes such as spray-and-wait scheme, we perform modeling study of two-hop single-copy scheme and two-hop multi-copy scheme under various transmission scheduling schemes, via queuing network analysis and continuous time Markov Chain model analysis. Our modeling analysis provides insights into the impact of source counter on routing performance and further suggests the existence of optimal counter value. Relying on the insights gained via simulations and modeling studies, we propose an adaptive scheme where nodes adjust their counter values to achieve minimum packet delivery delay, in a distributed and asynchronous fashion. Simulations demonstrate the effectiveness of our scheme and suggest the potential of exploring this rich area for improving DTN routing performance. 1. Introduction For Disruption Tolerant Networks, i.e., sparse and/or highly mobile networks in which there may not be a contemporaneous path from source to destination, routing Corresponding author addresses: xzhang@fordham.edu (Xiaolan Zhang), zhang@mcs.suffolk.edu (Honggang Zhang),yugu@nec-labs.com (Yu Gu) Preprint submitted to Elsevier October 8, 2010

2 protocols typically adopt a store-carry-forward paradigm, where each node in the network stores a copy of the data packet that it generates or that have been forwarded or duplicated to it by other nodes, carries the packet while it moves around in the network, and forwards or duplicates the packet to other nodes (or the destination node) when they come within transmission range [24, 13]. DTN routing consists of two fundamental components: the overall packet routing scheme chosen by the source node; and the transmission scheduling scheme adopted by each node for the data packets in its local buffer. The routing scheme chosen by the source node mainly decides the number of replicates allowed for each packet in the network and the number of hops that a packet is allowed to travel in the network. The transmission scheduling scheme adopted by a node decides which packets in its local buffer to forward when it encounters another node. Most of the early proposed DTN routing schemes assumed that there was no resource limitation in the network, and therefore focused mainly on the routing scheme aspect. Routing schemes can be classified into single-copy or multi-copy schemes. Under a single-copy scheme, at any time, there is at most one copy of each packet in the network. In other words, each packet is forwarded along a single path from source to destination. Under a multi-copy scheme, multiple copies of a packet are allowed to concurrently travel in the network; a packet can be copied (i.e., duplicated) to other nodes, allowing simultaneous use of multiple paths to the destination. For example, epidemic routing [24] scheme allows a node carrying a copy of a particular packet to duplicate it to every other node it encounters (if the other node does not have a copy yet). Variations of epidemic-like routing schemes include spray-and-wait [23, 21, 22], K-hop and probabilistic forwarding [11]. Compared to single-copy schemes, multi-copy schemes enjoy better delivery performance (i.e., lower delivery delay and higher delivery probability), at the expense of higher transmission overhead and buffer occupancy. For a classification of DTN routing schemes, interested readers are referred to [2]. More recently, several works (such as [3, 2, 16]) addressed routing in DTNs that are subject to resource constraints. In practice, transmission bandwidth in DTNs is often limited due to the low data rates of wireless radio and the short duration of node-to-node encounters. Applications such as mobile sensor networks often deploy small batterypowered nodes, hence energy and memory capacity are also scarce resources. Research in [3, 2, 16] addressed the transmission scheduling (and buffer management, if the storage is constrained) problems for resource constrained DTNs, assuming that the routing scheme chosen by source nodes are fixed (e.g., epidemic routing is often employed). And they (e.g., [2, 16]) mainly rely on heuristics for improving routing performance which imposes lots of control traffic for information exchange, as Balasubramanian et al [2] shows that finding an optimal schedule for DTN routing is NP-hard. In this paper, we study how to optimize the routing performance in DTNs where transmission bandwidth is scarce resource but power and storage are not constrained (e.g., vehicle based DTNs [3, 26, 7]). Since bandwidth is limited, thus it is critical to schedule the routing and transmission of packets optimally for achieving optimal routing performance. In contrast to existing works, we explore an alternative way to improve DTN routing performance via data source control, assuming the transmission scheduling of local buffer of each node is given and fixed. By data source control, we mean that a source node (at the application or transport layer) decides to choose a 2

3 particular routing scheme such as epidemic routing, or a spray-and-wait scheme with certain counter number for each packet. Here we assume that all nodes in a network are fully cooperative in carrying out the routing scheme chosen by a source node. We study the spray-and-wait counter-based routing scheme, with focus on the 2-hop multi-copy scheme. The central question we address in this paper is: can source nodes improve their routing performance by adjusting the duplication factor for each packet? Our main contributions are summarized as follows. First, we observe via simulations that there exists an optimal counter value that achieves the minimum average network-wide packet delivery delay. Then as a first step towards understanding multihop multi-copy DTN routing schemes, we model a two-hop multi-copy DTN routing via a continuous time Markov Chain. This modeling analysis provides insights into the impact of counter on routing performance and further suggests the existence of optimal counter value. In this process, we study the capacity region of DTN routing, and accurately analyze the average packet delivery delay under the two-hop single copy relaying scheme. Relying on the insights gained via simulations and modeling, we design an adaptive scheme that allow nodes to adaptively adjust their counter values (in search for an optimal counter value) to achieve minimum packet delivery delay. Our simulation studies demonstrate the effectiveness of our scheme and suggest the great potential of exploring this approach to improve DTN routing performance. This paper extends and improves upon our earlier workshop paper [28] in the following ways. We extend the analysis of the two-hop single-copy scheme. We have generalized the equal allocation scheme to the proportional allocation scheduling scheme. We have also performed an in-depth analysis of both proportional allocation and priority scheduling scheme where we discuss the conditions for the schemes to be stable and provide an explanation to the modeling errors observed for the priority scheduling scheme. For the two-hop multi-copy scheme, we have clarified the modeling of IMMUNE recovery scheme in the model, and presented more technical details for the numerical solution of the model. We have also performed a comparison of all routing schemes considered in this paper. The remainder of this paper is organized as follows. In Section 2, we discuss related work. Section 3 presents the network and traffic model considered in this work, and discusses the maximum network throughput. Then in Section 4 we present our simulation results that show the existence of an optimal counter for the spray-and-wait routing scheme. In Section 5, we present a queuing system analysis of the two-hop single copy routing scheme, considering both proportional allocation and priority scheduling. In Section 6, we propose a Markov Chain model for two-hop K-copy routing, and present the modeling studies and simulation studies that explore the impact of the protocol parameters and compare the performance of different routing schemes. In Section 7, we demonstrate the effectiveness of counter-adaptation routing control based on our modeling results. Finally, we conclude our paper in Section Related Work For DTNs without resource constraint, there exists a fundamental trade-off between the routing performance (in terms of delivery delay and delivery ratio) and overhead [27]. Various research works explored this trade-off in their studies of DTN routing 3

4 schemes. For example, [19, 1] proposed the optimal probabilistic routing schemes with the goal of achieve optimal tradeoff between delay and power, or achieve optimal delivery probability while satisfying certain power constraint. For mobile sensor networks, [25] proposed data delivery schemes that achieve the desired data delivery ratio with minimum overhead (i.e., power consumption). DTNs with bandwidth and/or buffer constraints have drawn more attention from researchers in recent years. [2, 16] have taken resource constraints into consideration when designing routing schemes. While these works focused on the design of the local transmission scheduling and buffer management policy adopted by the network nodes in order to optimize system wide performance such as average packet delivery delay, we focus on a different aspect of DTN routing control, i.e., the choice of duplication factor at the source node. A common assumption made by [2, 16] and our paper is that all network nodes are cooperative in optimizing network wide performance. In [20, 5], incentive mechanisms for fostering cooperation in DTNs with selfish nodes were proposed. As to the performance modeling of DTN routing schemes, the majority of existing work (e.g.,[9, 27, 12]) assume there is no bandwidth constraint, i.e., when two nodes meet, they can transfer an unlimited number of packets. To the best of our knowledge, [14] is the only modeling work taking into account bandwidth constraint. [14] studies delivery delay under epidemic routing and spray-and-wait routing when purely randomized scheduling scheme is adopted by the network relay nodes. We however focus on the two-hop, single-copy and multi-copy routing scheme, and take into considerations more sophisticated scheduling schemes. In particular, we study proportional allocation of bandwidth between source-to-relay and relay-to-destination transmission, and prioritized scheduling scheme where the relay-destination transmission is given strictly higher priority than the source-relay transmission, For packets of same type of transmission, we study First-Come-First-Serve (FCFS) scheduling (for single-copy scheme) and randomized scheduling scheme (for multiple-copy scheme). As [2, 16] pointed out and our simulation studies confirmed, prioritized scheduling outperforms purely randomized scheduling scheme. [17] characterized the scaling properties (i.e., asymptotic behavior) of the throughput and delay of DTNs under the two-hop relay scheme proposed by Grossglauser and Tse [10]. Based on a different DTN model, we have performed an accurate analysis on the delivery delay under the two-hop relay scheme (with considerations for more transmission scheduling schemes), and demonstrated that this scheme achieves the network capacity for DTNs with even number of nodes and circular traffic pattern. 3. System Model In this section, we first present the network and traffic model considered in this paper. We then discuss the maximum per-flow throughput that can be supported by the network. Finally, we introduce the spray-and-wait scheme and the two-hop singlecopy/multi-copy scheme, together with the different transmission scheduling schemes studied in this paper. 4

5 Notation N β B λ K P SR P RD T d T p E[C r] c min c inc c dec Description number of nodes in the network pair-wise inter-meeting rate number of packets that can be exchanged during a contact per flow packet generating rate packet duplication factor used in two-hop K-copy scheme fraction of bandwidth used for source-relay transmission fraction of bandwidth used for relay-destination transmission packet delivery delay packet propagation duration average number of relay nodes carrying a copy of a packet the minimal spray-and-wait counter to use the increment step when adapting spray-and-wait counter the decrement step when adapting spray-and-wait counter Table 1: Summary of Notations 3.1. Network and Traffic Model Consider a network of N mobile nodes moving within a closed region. Each node has a limited transmission range such that the network is sparse and disconnected. Let inter-meeting time between a pair of nodes denote the duration of time from the time when the two nodes move out of transmission range of each other to the next time they come into range of each other. We assume in this paper that the inter-meeting time between each pair of nodes follows an exponential distribution with rate β, an assumption made by most previous modeling work in DTNs (e.g.,[9, 27, 12]). This assumption is motivated by the findings in [9] which showed that under random waypoint and random direction models, the pair-wise inter-meeting time follows an exponential distribution when node velocity is relatively high compared to the region size and the transmission range is relatively small. Furthermore, [4, 15] show that under a large class of mobility scenarios, the inter-meeting time follows a power-law up to a point, and then exhibits an exponential decay. As far as we know, [17] is the only analytic work that is based on such two-phase distributed inter-meeting time model where only asymptotic results for the two-hop single-copy scheme are obtained. We consider bandwidth constraint by assuming a total of B packets can be exchanged between two nodes during their encounter. Note that the unit of bandwidth B can be interpreted in many different ways: B = 2 can represent that two packets or messages or files are exchanged per nodal contact. Without loss of generality, we will use packets per contact as B s unit. We consider the following traffic model commonly adopted in DTN and MANET literature ([10, 27, 17]). There are N unicast flows in the network, with packets arriving at each source node (and flow) according to a Poisson process with rate λ. Each node is the source of only one flow, and the destination of another (and only one) flow. Let f i denote the unicast flow originated from node i (1 i N), and d(f i ) denote the destination node of flow f i. We have d(f i ) {1, 2,..., N} {i} and d(f i ) d(f j ) for i j, 1 i, j N. Under this traffic model, an equal amount of traffic is 5

6 min cut 3 4 (a) A circular traffic pattern for N = 6 case. This figure only shows the traffic flows (b) Min cut for the network shown in (a). This figure only shows the traffic flows. There are 9 edges cross the cut. 6 Figure 1: Example of a network with even number of nodes and circular traffic pattern originated at and destined to each node in the network. As each node is the source of one flow (the destination of another flow), under the two-hop routing schemes, there is no inter-flow scheduling issue, i.e., selecting from multiple flows during the sourceto-relay transmissions (relay-to-destination transmissions). Such homogeneous traffic model is a reasonable assumption for modeling studies, given the lack of application traffic models that are derived from real DTN traces. As we focus on DTN scenarios where the nodes are not power constrained, the performance metric of interest is the delivery performance, more specifically, the packet delivery delay, defined as the time duration from when a packet is generated at the source node to the time when the packet is first delivered to the destination. Our systemwide optimization goal is to minimize the average delivery delay of all packets from all flows. The simulations reported in the paper were carried out using a discrete event DTN simulator that we developed. We simulated the node mobility by drawing the pair-wise inter-meeting time from an exponential distribution with rate β. For each simulation run, the N unicast flows are configured randomly as follows: for each node, we choose a node uniformly at random (from all nodes that have no flow destined to it yet) to be the destination of its generated packets. Simulation setting can be described as a 4-tuple (N, β, B, λ). N, β, B, and λ, together with other notations used in the paper, are summarized in Table 1. Throughout the paper, we report average packet delivery delay for packets generated after the system has entered a steady state. The confidence interval of the average delivery delay is not reported as the interval is very small due to the large sample size (i.e., a large number of delivery delay samples) used Maximum Network Throughput Before presenting DTN routing schemes studied in this paper, we first consider the following question: for a network setting given by a 3-tuple (N, β, B), what is the maximum per-flow traffic rate that can be supported 1? The maximum per-flow 1 that is, there exists a routing scheme that can sustain the per-flow traffic rate. 6

7 traffic rate is an important reference point for both the modeling analysis and simulation studies. When evaluating the maximum network throughput, a DTN can be viewed as a static undirected network where every node is connected with every other node in the network, and the long-term average bandwidth of each link is Bβ. The maximum achievable per-flow throughput can be found by solving a classical maximum concurrent multi-commodity flow problem, which in general can be solved as a Linear Programming problem [6]. Nevertheless, as demonstrated in Section 5.1, the two-hop single copy schemes can support a per-flow throughput of λ = NBβ/4, therefore providing a lower bound on the maximum achievable per-flow throughput. For the special case where the network has an even number of nodes and a circular traffic pattern (e.g., Fig 1.(a)), we obtain an upper bound on the maximum achievable per-flow throughput as follows. In [18], it has been shown that the min-cut (i.e., sparsest cut) of a (undirected) multi-commodity flow problem is an upper bound of the max-flow, where the min-cut is defined to be the cut with the minimum ratio (among all cuts) of the cut capacity over the demand over the cut. Fig 1.(b) illustrates the min-cut for the multi-commodity flow problem shown in Fig 1.(a). For all such networks, i.e., with an even number of nodes and a circular traffic pattern, we form the min-cut by dividing the set of N nodes, V, into two sets U and V U, each with N/2 nodes, such that for each of the N unicast flows, its source node is in one set and its destination is in another set. The capacity of the cut is N/2 N/2 Bβ, obtained by multiplying the number of links crossing the cut with the link bandwidth. The demand of the cut, i.e., the number of unicast sessions flowing through the cut, is N. The value of this cut is therefore NBβ/4, and one can easily show that this cut is the min-cut. We therefore conclude that the maximum per-flow throughput of the network is upper-bounded by λ = NBβ/4. Furthermore, as the two-hop single-copy schemes (shown in Section 5.1) can support a per-flow throughput of λ, we conclude that for this particular network setting, i.e., a DTN with an even number of nodes forming a circular traffic pattern, the maximum per-flow throughput is λ DTN Routing Schemes In this section, we present the routing schemes studied in this paper. We start with the spray-and-wait scheme which is an multi-hop multi-copy routing scheme. We then describe the two-hop single-copy scheme and two-hop multi-copy schemes, together with the transmission scheduling schemes considered in this paper Spray-and-Wait Scheme Under binary spray-and-wait [23, 21, 22] scheme with a counter value of L, the source node assigns a token value of L to the source packets it generates. When a node u carrying the packet with n(n > 1) tokens meets a relay node v that does not carry a copy, node u copies the packet to node v and split the n tokens in half with node v. When a node has one token left, it only delivers the packet to the destination (i.e., no more replication). It has been shown that under an independent and identically distributed mobility model, binary spray-and-wait scheme minimizes the expected time to distribute all L copies [22], and therefore minimizes the packet delivery delay. 7

8 Two-Hop Single-Copy Scheme In their seminal work [10] that demonstrated the positive effect of mobility on throughput, Grossglauser and Tse proposed a two-hop single-copy scheme that leverage the mobility of nodes in a mobile ad hoc network to deliver packets via a one-hop or two-hop path. Under this scheme, the source node either directly transmits its source packet to the the destination when they come into transmission range of each other, or forwards the packet to one of the N 2 non-destination nodes (relay nodes) which stores the packet and delivers the packet to the destination when they encounter each other. In particular, when node i and j encounter each other, the total bandwidth B are equally allocated to packet transmission in the two directions (from node i to j, and from node j to i), i.e., B/2 packets can be transferred in each direction. We now consider the packet transmission from node i to j (the packet transmission from node j to i is similar). If node j is the destination node of flow f i, i.e., j = d(f i ), then the only possible type of transmission from node i to j is the direct source-destination transmission to which all bandwidth (B/2 packets per contact) is allocated. If node j is not the destination node of flow f i, then j acts as a relay node for flow f i, and node i acts as a relay node for the flow destined to j. [10] considered an equal allocation scheme, where the total available (one-way) bandwidth, B/2, is allocated as follows, i.e., B/4 packets for source-relay transmission from source node i to relay node j, and B/4 packets for relay-destination transmission from relay node i to destination node j. Packets of same transmission type are served (i.e., transmitted) in the First-Come- First-Serve order. In this paper, we not only generalize the equal allocation to proportional allocation, but also consider priority scheduling schemes. Under the proportional allocation, the total one-way bandwidth (B/2 packets per contact) is allocated to the relayto-destination transmission and the source-to-relay transmission in proportion, e.g., P RD (0 < P RD < 1) fraction of the bandwidth is allocated to relay-to-destination transmission, while P SR (which equals to 1 P RD ) fraction of the bandwidth is allocated to source-to-relay transmission. Motivated by those works that give higher priority to packets that are destined to the receiver node [2, 16], the priority scheduling scheme gives strictly higher priority to relay-to-destination transmission over sourceto-relay transmission. More specifically, for the transmission from node i to j, where j d(f i ), node i first transmits all relay packets that are destined to node j, and then transmits its own source packets to node j (which acts as a relay) if there is remaining bandwidth Two-Hop K-Copy Scheme Under the two-hop K-copy scheme, the source node replicates a source packet to up to K (K 1) relay nodes; each of the relay nodes can only forward the packet to the destination. At any point of time, there might be multiple copies of a packet in the network, including the copy carried by the source node and up to K copies in the relay nodes. The packet is first delivered to the destination when a node carrying a copy of the packet encounters the destination that does not have the packet and chooses to forward the packet. Note that under the two-hop 1-copy (i.e., K = 1) scheme, the source node duplicates the packet to 1 relay node, resulting in up to 2 copies of each 8

9 packet in the network at any time. This distinguishes it from the two-hop single-copy scheme introduced in the previous section. The modeling study of the performance of two-hop K-copy scheme is much more challenging than the single-copy scheme, and we focus on the proportional allocation with randomized scheduling scheme for tractability. As before, when two nodes, i and j, encounter each other, the total bandwidth B is equally allocated to transmission in the two opposite directions. The packet transmission from node i to j proceeds as follows. If node j is the destination of flow f i, all available bandwidth (B/2) is allocated to source-destination transmission. Otherwise, node i schedules source-relay transmission with probability P SR, and schedules relay-destination transmission with probability P RD (= 1 P SR ). For each type of transmission, purely randomized scheduling is used to pick a packet from all eligible packets to transmit. Being a multi-copy scheme, the two-hop K-copy scheme employs control signaling to avoid transmitting a packet to a node multiple times. When two nodes, i and j, encounter each other, they first exchange signaling messages which contain the IDs of of the packets that they carry, and then choose packets (based on the transmission scheduling scheme) from all packets of which the other node does not have a copy. For multi-copy schemes, recovery schemes [11, 27] have been proposed to save energy consumption. Different recovery schemes differ in the incurred signaling overhead and resource savings. For example, under the IMMUNE recovery scheme, when a node first delivers a packet to the destination, an anti-packet for the packet is generated and stored by the node and the destination. Subsequently, when the destination encounters other nodes carrying a copy of the packet, it transmits the anti-packet to those nodes which consequently discard their copies of the packet and store the anti-packet to avoid being re-infected by the packet in the future. More aggressive recovery schemes such as VACCINE scheme, where anti-packets are propagated by network nodes (not only the destination), improves routing performance at the expense of control signaling. We focus on IMMUNE recovery in this paper as the modeling of VACCINE recovery scheme increases the complexity of the model substantially. Although relatively simple compared to the spray-and-wait scheme with more complex scheduling scheme, the two-hop K-copy scheme has the following two important aspects of a DTN routing scheme and therefore understanding it is an important first step towards understanding more complexed DTN routing schemes. First, the choice of P SR (and P RD ) of a node reflects its level of cooperativeness. A node with a larger P SR is more selfish as it allocates more bandwidth to disseminate its own source packets, and less bandwidth to relay packets for other nodes. Second, a source node can adjust its intended usage of network bandwidth by adjusting the maximum number of copies to disseminate for each packet, i.e., K. 4. Impact of Counter on Routing Performance As we stated earlier, under limited link bandwidth, a key research issue is how to schedule the transmission of packets to optimize performance. Such a scheduling can be done by data source nodes and/or each node s transmission scheduling scheme. Existing routing schemes typically focus on individual nodes transmission scheduling and buffer management, assuming that the routing scheme chosen by source nodes are 9

10 220 Avg. Packet Delivery delay epidemic routing Spray-and-Wait Counter Figure 2: Average packet delivery delay under spray-and-wait scheme with varying counter, network setting: (N = 101, β = , B = 2, λ = 0.05). All nodes use same spray-and-wait counter. fixed. As there are theoretical and practical difficulties with the above approach, we pursue an alternative way in this paper to improve routing performance via routing control from source nodes. Specifically, we assume that the transmission scheduling of each node is given, and we explore whether source nodes can vary their counter values for packet replication such that the routing performance can be improved. To pursue this direction, we conduct a set of simulations of binary spray-andwait scheme and epidemic routing scheme under the network setting given by N = 101, β = , B = 2, λ = All nodes adopts same local transmission scheduling. Our simulation results plotted in Fig 2 shows that there exists an optimal counter value, denoted as L, which minimizes the average packet delivery delay. Using larger and smaller counter leads to larger delivery delay. Note that L depends on the system parameters. When the network load is very low, L can be very large and the spray-and-wait scheme approaches to epidemic routing. The above results motivate us to explore the potential of routing control by source nodes. We attempt to understand why the counter-based routing schemes have optimal counter values, and how to design simple yet effective algorithms to allow source nodes to improve their routing performance by adjusting packet replicating counters. Next, we start off with formally analyzing counter-based routing schemes. To keep our analysis tractable, we focus on two-hop routing schemes. We first analyze the two-hop single-copy scheme (Section 5), then we model the two-hop K-copy scheme using a continuous time Markov Chain (Section 6). 5. Two-Hop Single-Copy Schemes In this section, we study the two-hop single-copy schemes, analyzing the average packet delivery delay achieved under different transmission scheduling schemes. We assume that the bandwidth of each nodal contact is B = 2. This bandwidth is equally shared by transmissions in the two opposite directions, i.e., one packet can be transmitted in each direction during each contact. Each node has a queue for its own source packets, referred to as source queue, and N 2 queues, referred to as relay queues, each for a flow for which this node is a relay. 10

11 α 1 Relay Que 1 λ Src Queue α 1 Relay Que 2... α 1 Relay Que N 2 α 2 Figure 3: Queuing network model for Two-Hop Single-Copy schemes Note that this node is also the destination node of one flow in the network. Under the two-hop single-copy scheme, each packet is either delivered to the destination directly, or forwarded to a relay node which then delivers the packet to the destination. As illustrated in Fig 3, when a source packet is first generated at the source node, it is first queued in the source node s source queue. After a certain queuing delay, the packet is either directly delivered to the destination (with probability α 2 ), or forwarded (not duplicated) to a relay node (with probability α 1 ) where the packet is queued at the relay node s relay queue and forwarded to the destination later. We have (N 2)α 1 +α 2 = 1. The arrival and service rates of the source queue and relay queue, and the values of α 1 and α 2 depend on the transmission scheduling scheme employed. We assume that when a node meets the destination node of its source flow, all of the one-way bandwidth (B/2 = 1 packet per contact) is allocated to deliver packets in its source queue to the destination 2. If a node meets another node that is not the destination of its flow, we consider the following two scheduling schemes: proportional allocation scheduling where the total one-way bandwidth (B/2 = 1 packet per contact) is allocated so that the source-to-relay and relay-to-destination transmission is allocated P SR and P RD of the bandwidth, where P SR +P RD = 1, P SR > 0 and P RD > 0. priority scheduling, where relay-to-destination transmission is given strictly higher priority over source-to-relay transmission. We next present a queuing network analysis of the two-hop single-copy schemes with the above two scheduling schemes Proportional Allocation Scheduling We first consider the proportional allocation scheme. 2 Under the traffic pattern considered in this paper, there is no other type of transmission. 11

12 The source queue can be modeled as a M/M/1 queue. The arrival rate is per-flow packet arrival rate, λ. Each source packet is either directly delivered to the destination (when the source meets the destination) with a rate of β, or forwarded to a relay node when the source node meets one of the N 2 relay nodes. For a meeting between a pair of nodes that are not source-destination pair, P SR fraction of the bandwidth is allocated to the source-to-relay transmission, therefore the rate that a source packet is forwarded to some relay node is (N 2)βP SR. The service rate of the source queue is therefore β + (N 2)βP SR. The source queue is ergodic if and only if the arrival rate is smaller than the service rate, i.e., λ < β + (N 2)βP SR, (1) and has an average sojourn time (the total time a packet spent in the queue) of T s = 1 β+(n 2)βP. The probability that the source queue is empty, denoted as p SR λ 0, is given by p 0 = 1 λ β+(n 2)βP SR. 1 When a packet leaves the source queue, with probability α 2 = 1+(N 2)P SR, the packet is delivered to the destination, i.e., no further delay; with probability 1 α 2 the packet is forwarded to one of the N 2 relay nodes. The probability that the packet is P SR 1+(N 2)P SR. forwarded to any particular relay node is α 1 = 1 α2 N 2 = Each relay queue at a node can also be modeled as an M/M/1 queue. Packet arrives whenever the relay node meets the source of the flow with a non-empty source queue and source-to-relay transmission is scheduled. The arrival rate to the relay queue is then (1 p 0 )βp SR (only P SR fraction of the meetings are used for source-to-relay transmission). The service rate of the queue is βp RD, considering that P RD fraction of the meetings between the relay to destination node are used for relay-to-destination transmission. The relay queue is ergodic if and only if (1 p 0 )βp SR < βp RD, (2) and the sojourn time is T r = 1 βp RD βp SR+p oβp SR. In summary, the average packet delivery delay under the two-hop single-copy scheme with proportional allocation is: T d = T s + (N 2)P SR 1 + (N 2)P SR T r. (3) For a given network setting specified by the tuple (N, β, λ, B = 2), the value chosen for P SR decides whether the scheme can support the offered traffic rate so that the network is stable 3. We solve inequalities (1) and (2) to obtain the range of values that P SR can take in order for the scheme to be stable. Fig 5.(a) plots the value ranges for the P SR under varying per packet arrival rate, λ. We observe that when λ is small, P SR can take a larger range of value; and when λ approaches λ, the upper bound and lower bound intersect at the value of In the sense that all queues in the network have bounded time average backlog [8]. 12

13 P SR Lower bound based on Inequality (1) Upper bound based on Inequality (2) Per Flow Packet Arrival Rate Average Delivery delay Simulation result Model prediction Bandwidth allocated to Source-Relay Transmission, P SR Average Delivery delay Simulation result Model prediction Per Flow Packet Arrival Rate (a) Constraints on P SR (b) Delivery delay under varying P SR, λ = 0.1 (c) Equal Allocation (P SR = 0.5) under varying λ Figure 4: Two-hop single-copy relay scheme with proportional allocation, N = 101, β = , B = 2, λ = NBβ 2 = Furthermore, within the above identified value range for P SR, the value chosen for P SR has a significant impact on the system wide packet delivery delay. Fig 5.(b) plots the average packet delivery delay (both model prediction and simulation result) under varying P SR for the case with λ = 0.1, where the value range for P SR is [0.196, ] according to Fig 5.(a). We observed a perfect match between the model prediction and simulation result. We also observe that as P SR increases, the average packet delivery delay first decreases, and then increases. This is because as Eq. (3) shows, the average packet delivery delay consists of two parts, the queuing delay at the source queue, T s, and the queuing delay at the relay queue, T r. As P SR increases, T s decreases, T r increases and the probability of experiencing relay queue delay increases too, leading to the observed behavior of packet delivery delay. For a given network setting, there is a value for P SR that minimizes the average packet delivery delay, denoted as PSR = argmin P SR T d. We can obtain PSR by solving the following equation: dt d dp SR = 0, for P SR. From Fig 5.(a), we also observe that when P SR = 0.5, as long as λ < λ, both inequalities (1) and (2) are satisfied and therefore the scheme is stable. This scheme, i.e., with P SR = 0.5, is exactly the two-hop single-copy scheme proposed in [10]. Interestingly, under such equal allocation, the source queue and relay queue have same utilization factor (ρ = 2λ Nβ ) and average queue length. Both queues are stable as long as λ < Nβ 2 = λ, therefore the scheme can support a per-flow throughput of λ 4. The following closed form expression for average packet delivery delay can be obtained from Eq. (3): T d = T s + (1 2/N)T r = 2N 2 Nβ 2λ. Fig 5(c) plots the average packet delivery delay under varying per-flow traffic rate, comparing the simulation results against the model prediction. The model proposed 4 This result can be extended to the case where B > 2. 13

14 in this section accurately predicts the average packet delivery delay for the two-hop single-copy scheme under proportional scheduling scheme Priority Scheduling The proportional allocation scheme considered in the previous section is more tractable for analysis, as the fixed proportional allocation allows us to obtain the arrival and service rates of the source queue and relay queue. However, as [2, 16] have demonstrated, higher priority should be given to delivery traffic (including source-destination and relay-destination). In this section, we analyze the two-hop single-copy scheme with priority scheduling where relay-to-destination transmission is given strictly higher priority than source-to-relay transmission. Recall that a queuing network model as depicted in Fig 3 can be used to model a two-hop single-copy scheme. For a node in the network, we denote by N S the number of packets in its source queue and N R the number of packets in each relay queue (there are N 2 relay queues in each node), and let r 0 = Pr{N R = 0} and s 0 = Pr{N S = 0}. We first consider the arrival and service rate of the source queue. Source packets arrive to the source queue with packet arrival rate, λ. A source packet leaves the queue when the source directly delivers the packet to the destination (with rate Bβ/2 = β), or when the source forwards it to a relay node. Recall that source-to-relay transmission has lower priority than relay-to-destination traffic: only when the source node has no relay packet destined to the receiver, will the source packet be forwarded. Therefore, the source node forwards a source packet to a relay node with rate βr 0, and the rate that a source packet is forwarded to one of the N 2 relay nodes is: (N 2)βr 0. The total service rate of source queue is therefore β + (N 2)βr 0. The condition for the source queue to be stable is: We also have s 0 = Pr{N S = 0} = 1 and the average sojourn time of the source queue is T s = λ < (N 2)βr 0. (4) 1 β + (N 2)βr 0 λ. λ β + (N 2)βr 0, (5) After a source packet leaves the source queue, with probability (N 2)βr0 β+(n 2)βr 0, the packet enters some relay queue, where it experiences another delay before being forwarded to the destination, i.e., α 1 = 0 β βr β+(n 2)βr 0, and α 2 = β+(n 2)βr 0. For the relay queue, a packet arrives when the relay node meets the source with a non-empty source queue, and the source carries no relay packet that is destined to the relay node. Therefore the packet arrival rate to the relay queue is βpr{n S 0&&N R = 0} βpr{n S 0}Pr{N R = 0} = β(1 s 0 )r 0, (6) 14

15 assuming that N S 0 (i.e., the node has a non-empty source queue) is independent from N R = 0 (i.e., the node has an empty relay queue for the node it meets). The service rate of the relay queue is β, as the relay-to-destination traffic has strictly higher priority over source-to-relay traffic. No matter what is the value of λ, the relay queue is always stable as its arrival rate is always smaller than the service rate. For this M/M/1 queue, we have which yields r 0 = Pr{N R = 0} = 1 β(1 s 0)r 0, β The sojourn time of the relay queue is then given by T r = r 0 = 1/(2 s 0 ). (7) 1 β β(1 s 0 )r 0. In summary, the average packet delivery delay is: T d = T s + (N 2)βr 0 β + (N 2)βr 0 T r, (8) where r 0 and s 0 can be calculated by solving the equations Eq.(5) and Eq.(7). We plug the value of r 0 to inequality (4) and find the maximum per packet flow rate that can be supported by this scheme. In Fig 5.(a), the arrival and service rate of the source queue under varying per-flow packet arrival rate is plotted, and we observe that as long as the per packet arrival rate is smaller than λ = , the source queues are stable, and so is the scheme. Fig 5.(b) compares the model predicted average delay with the simulation result which shows a very good match for λ < 0.15, and an underestimation of delivery delay for heavier traffic rate. This is due to the approximation in Eq (6) based on the assumption of the independence between the events of a non-empty source queue and an empty relay queue during a source-relay encounter. Our simulation studies indeed show a mismatch between Pr{N S 0&&N R = 0} and Pr{N S 0}Pr{N R = 0}, especially for heavy load case. 6. Modeling Studies of Two-Hop K-Copy Scheme When the per-flow traffic rate is low, the available network bandwidth is not fully utilized by the two-hop single-copy schemes. To reduce packet delivery delay, a packet can be replicated in the network and therefore multiple copies of the packet are propagated simultaneously through multiple paths. In this section, we analyze the two-hop K-copy scheme for the homogeneous case where all nodes use the same P SR. We propose a continuous time Markov Chain (MC) to model a packet s lifetime in the network, and then numerically solve the MC, coupled with queuing analysis of the source queue and relay queue, to obtain average packet delivery delay. 15

16 Source Que Arrival and Departure Rate λ= departure rate arrival rate Per Flow Packet Arrival Rate, λ Delivery delay Simulation result Model prediction Per Flow Packet Arrival Rate (a) Arrival & departure rate of source queue (b) Average Packet Delivery Delay Comparison Figure 5: Two-hop Single-copy scheme under priority scheduling, N = 101, β = , B = Markov Chain Model of A Packet s Life Time In this section, we present a Markov Chain that models the propagation and delivery of a typical packet in the network. For ease of explanation, we denote this packet as P. Suppose that packet P is generated at the source node at time t = 0, and the twohop K-copy (K 1) scheme is used to propagate the packet in the network where all nodes employ proportional allocation with randomized scheduling with same value for P RD. Fig 6 depicts the state diagram of the MC. Each state of the MC is denoted as (S I, R I, R), where S I denotes whether the source node has a copy of the packet (with value 1) or not (with value 0), R I denotes the number of infected or recovered relay nodes, and takes integer values ranging from 0 to K, and R denotes whether the packet has been delivered (with value 1) or not (with value 0). The initial state is (1, 0, 0), where only the source node carries a copy of the packet. The propagation of the packet stops when the source node recovers from the packet, i.e., it delivers the packet to the destination or meets the destination node that already received the packet 5. We hence use a single state (0,, 1) to represent all states where the source has recovered from the packet and there are different numbers of infected or recovered relay nodes. This MC is reducible and transient, where the state (0,, 1) is an absorbing state and the initial state (1, 0, 0) cannot be reached from any other state. There are five types of transitions in the MC, labeled as S-R, S-D, R-D and D-S respectively in the diagram. The S-D transition occurs when the source node meets the destination which has not received packet P yet, and from all eligible source-todestination packets, packet P is chosen to be transmitted. The S-R transition occurs when the source meets a susceptible relay node that does not carry a copy of packet P, and from all eligible source-relay packets, packet P is chosen to be transmitted. The R-D transition occurs when a relay node that carries packet P meets the destination that has not received the packet yet, and the relay node chooses packet P from the set of all eligible relay-destination packets. Finally, the D-S transition occurs when the destination that has already received packet P meets the source, and transmits the anti- 5 Remaining copies of the packet carried by relay nodes will be eventually deleted when the relay nodes meet the destination. 16

17 S D (1,0,0) (1,1,0) (1,2,0) (1,K 1,0) S D S D S D R D R D R D (1,K,0) R D S D (1,1,1) (1,2,1) (1,K 1,1) (1,K,1) D S D S D S D S (0,*,1) Figure 6: MC model of a packet s lifetime under two-hop K-copy scheme packet to the source. The source subsequently deletes its copy and stops propagating the packet. Note that under IMMUNE recovery, when the destination that has already received packet P meets one of the infected relay nodes, it transmits an anti-packet to the relay node allowing the relay node to become recovered (i.e., delete its copy). However, this does not result in a change in the MC state, due to the fact that the MC states record the sum of the number of infected and recovered relay nodes. Similar Markovian models of multi-copy DTN routing schemes such as epidemic routing and 2-hop routing have been proposed and studied in previous modeling studies [9, 11, 27, 14]. However, except [14], most works assume there are no resource constraint, and therefore each packet propagates in the network independently from other packets, allowing one to express the various transition rates more easily. To account for bandwidth constraint, we need to take into account the fact that when two nodes meet, there are multiple packets competing for the transmission opportunity. For the purely random scheduling, the probability that the packet P is chosen to be transmitted is equal to the reciprocal of the number of packets competing for the transmission opportunity. We now introduce three random variables that denote the number of packets competing for each of the three types of transmissions, i.e., source-to-destination, sourceto-relay, and relay-to-destination respectively. We will discuss how to evaluate them in Section 6.3. Let N SD denote the number of eligible source packets to be transmitted to the destination, when the source meets the destination, and E[N SD ] denote its expected value. And let N SR denote the number of source packets in the source that are eligible to be transmitted to a relay node when the two nodes meet, and E[N SR ] denote its expected value. Finally let N RD denote the number of relay packets that are eligible to be transmitted to the destination node when the two nodes meet, and E[N RD ] denote its expected value. We assume that these three quantities have a stationary distribution, which is true if the scheme stabilizes. With the expected values of the above three variables, we now analyze the transition rates of the MC: The S-D transition rate, i.e., the rate that the source node meets the destination which has not received packet P yet, and delivers packet P, is β/n SD, and can be approximated by β/e[n SD ]. Note that the source meets the destination node with a rate of β, and during such encounter, the source uniformly randomly 17

18 (1,0,0) (1,1,0) (1,2,0) (1,K 1,0) (1,K,0) S D R D, S D R D,S D R D,S D R D,S D D Figure 7: MC model of a packet s lifetime until delivery chooses a packet from the set of all eligible source packets (there are N SD such packets) to transmit, therefore the probability that packet P is chosen is 1/N SD, which is approximated by 1/E[N SD ]. That S-R transition rate, i.e., the rate that the source meets a susceptible relay nodes (i.e., relay nodes that do not carry a copy of or an anti-packet for packet P ), and transmits packet P to the latter, is state dependent. If there are currently R I relay nodes that either carries a copy of packet P or carries an anti-packet for packet P, then the S-R transition rate is (N 2 R I )P SR β/e[n SR ]. This is because the source meets a susceptible relay node with rate (N 2 R I )β (note that there are N 2 R I susceptible relay nodes in the network), with probability P SR /N SR (which we approximate by P SR /E[N SR ]), the source performs source-to-relay transmission and chooses packet P to transmit to the relay node encountered. The R-D transition rate is the total rate that one of the relay nodes that carry packet P meets the destination that has not received the packet yet, and deliver packet P to the destination. The total encountering rate between infected relay nodes and the destination is R I β (given that the destination has not received the packet yet, the number of infected relay nodes is R I, i.e., there is no recovered relay node.). With probability P RD, the encountering performs relay-destination transmission. The probability that packet P is chosen to transmit, under uniformly random scheduling, is 1/N RD. Therefore, the R-D transition rate can be approximated by R I βp RD /E[N RD ]. The D-S transition rate is the rate that the destination which has already received packet P meets the source. As we assume there is no bandwidth constraint for control signaling, the D-S transition rate is β, the rate that the source and the destination encounter each other Packet Delivery Delay, Propagation Duration and Number of Copies in Network We now derive average packet delivery delay (E[T d ]), average packet propagation duration (E[T p ]), and average number of copies in the network (E[C r ]) from the MC model. The packet delivery delay is of interests in its own right, where the other two metrics are needed for estimating E[N SD ], E[N SR ] and E[N RD ] as the next section details. 18